LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – MATHEMATICS
THIRD SEMESTER – APRIL 2012
MT 3811 – COMPLEX ANALYSIS
Date : 24-04-2012 Dept. No. Max. : 100 Marks
Time : 1:00 – 4:00
Answer all the questions.
- a) Prove that if using Leibniz’s rule
- b) Let be a non-constant polynomial. Prove that there is a complex number such that. (5)
- c) Let be an analytic function. Prove that for where Hence prove that if f is analytic in an open disk then prove that for where.
- d) State and prove homotopic version of Cauchy’s theorem. (15)
- a) State and prove Morera’s theorem.
- b) Prove that a differentiable function on is convex if and only if is
- c) State and prove the Arzela-Ascoli theorem.
- d) State and prove the Riemann mapping theorem.
- a) Show that in the usual notation.
- b) If and then prove that .
- c) (i) Let be a compact metric space and let be a sequence of continuous functions from X into such that converges absolutely and uniformly for x in X. Then prove that the product converges absolutely and uniformly for x in X. Also prove that there is an integer such that if and only if for some n, .
(ii) State and prove Weierstrass factorization theorem. (7+8)
- d) Let , then prove that converges absolutely if and only if converges absolutely.
- e) State and prove Bohr-Mollerup theorem. (7+8)
- a) State and prove Jensen’s formula.
- b) If is a metric space, then prove that is also a metric on (5)
- c) State and prove Rung’s theorem. (15)
- d) State and prove Hadamard’s factorization theorem. (15)
- a) Prove that any two bases of a same module are connected by a unimodular transformation.
- b) Prove that an elliptic function without poles is a constant.
- c) (i) Prove that the zeros and poles of an elliptic function satisfy .
(ii) Derive Legendre’s relation (7+8)
- d) (i) State and prove the addition theorem for the Weierstrass
(ii) Show that